Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mapssfset | Structured version Visualization version GIF version |
Description: The value of the set exponentiation (𝐵 ↑m 𝐴) is a subset of the class of functions from 𝐴 to 𝐵. (Contributed by AV, 10-Aug-2024.) |
Ref | Expression |
---|---|
mapssfset | ⊢ (𝐵 ↑m 𝐴) ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapfset 8445 | . . 3 ⊢ (𝐵 ∈ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑m 𝐴)) | |
2 | eqimss2 3951 | . . 3 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑m 𝐴) → (𝐵 ↑m 𝐴) ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐵 ∈ V → (𝐵 ↑m 𝐴) ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}) |
4 | reldmmap 8431 | . . . 4 ⊢ Rel dom ↑m | |
5 | 4 | ovprc1 7195 | . . 3 ⊢ (¬ 𝐵 ∈ V → (𝐵 ↑m 𝐴) = ∅) |
6 | 0ss 4295 | . . 3 ⊢ ∅ ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵} | |
7 | 5, 6 | eqsstrdi 3948 | . 2 ⊢ (¬ 𝐵 ∈ V → (𝐵 ↑m 𝐴) ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}) |
8 | 3, 7 | pm2.61i 185 | 1 ⊢ (𝐵 ↑m 𝐴) ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 {cab 2735 Vcvv 3409 ⊆ wss 3860 ∅c0 4227 ⟶wf 6336 (class class class)co 7156 ↑m cmap 8422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-id 5434 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-fv 6348 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7699 df-2nd 7700 df-map 8424 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |